Partly a rant from today’s New York Times

 source—note the “\cdots”

Aryabhata is often called the father of Indian mathematics, and can be regarded as the progenitor of many modern features of mathematics. He completed his short great textbook, the Aryabhatiya, in 499 at age 23. Thus if there had been a “STOC 500,” he could have been the keynote speaker. The book has the first recorded algorithm for implementing the Chinese Remainder Theorem, which Sun Tzu had stated without proof or procedure. He computed ${\pi}$ to 4 decimal places, and interestingly, hinted by his choice of words that ${\pi}$ could only be “approached.” He introduced the concept of versine, that is ${1 - cos(\theta)}$, and computed trigonometric tables. He maintained that the Earth rotates on an axis, 1,000 years before Nicolaus Copernicus did, and while he described a geocentric system, he treated planetary periods in relation to the Sun.

Today Ken and I wish everyone a Happy Father’s Day, and talk about recognizing our scientific fathers correctly. This includes a short rant about a flawed piece in today’s New York Times.

Ken believes that Aryabhata further hinted at the “correct” value of ${\pi}$. As earlier referenced by our fellow blogger and friend Bill Gasarch, this article by Bob Palais wins Ken’s assent that we should really think in terms of 6.283185307… Here is what Aryabhata wrote:

Add 4 to 100, multiply by 8, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached.

Why wouldn’t he say to multiply by 4, add 31,000, and get the circumference for a diameter of 10,000? Ken observes that 10,000 was already the more recognizable unit, so Aryabhata may have been hinting at the ratio to the radius. Indeed Aryabhata flatly stated a decimal place system as the norm for arithmetic, as already used in earlier manuscripts he cited, though he himself extended it with a system of Sanskrit phonemes capable of handling fractions and decimal places.

Aryabhata is sometimes also credited with the first symbolic use of zero, at least implicitly. His successor Brahmagupta, a century-plus later, perhaps deserves greater credit for being the first to treat zero as an explicit entity. He wrote rules such as zero plus anything is itself, zero times anything is zero, and “a fortune subtracted from zero is a debt.” Brahmagupta waded into trouble only when he ruled that zero divided by zero is zero, and tried to make unique sense out of other fractions with zero. But our point is that in the West he seems to get zero credit for zero at all. And that brings me to giving credit to “fathers” of fields correctly.

## Today’s Article

This Sunday’s New York Times has, on page nine of the Review section, an article by Ignacio Palacios-Huerta, which is titled “The Beautiful Data Set.” He is a professor of economics at the London School of Economics, and is the author of a recent book on Game Theory.

In his article he uses soccer as a way to explain game theory. He is especially interested in penalty kicks. These are two person games between the goalie and the player. It is a zero-sum game, since either the kick goes in or it does not. He has looked at over nine thousand such kicks in professional games that took place over the period 1995 to 2012. He found that roughly 60% go to the right of the net and the rest to the left. This is, as he notes, not exactly an even distribution; this is probably due to players having a stronger kick to one side than the other.

All of this is interesting, and I like when the NYT, or any mainstream media, talks about mathematics. But Palacios-Huerta repeatedly gets one thing wrong. I cannot understand why, but in my opinion he does. He repeatedly attributes the theory of such zero-sum games to John Nash. He calls all of the above “Nash’s Theory.”

This is just wrong. The theory of zero-sum games is due to John von Neumann. Recall that he published the famous Minmax Theorem in 1928 and said:

As far as I can see, there could be no theory of games … without that theorem … I thought there was nothing worth publishing until the Minimax Theorem was proved.

Palacios-Huerta knows this—he quotes the above in his book—so I am at best puzzled by his use of Nash in his NYT article. The only point I can see is that the public may recall Nash, since he won a Nobel Prize and is featured in a great book and even a major motion picture. Soccer is after all called the “Beautiful Game,” whence his “Beautiful Data Set,” and the association with “A Beautiful Mind” called up Nash to either him or the editor—or both.

But I still am disappointed with the NYT. The whole claim to fame for Nash is that he extended the theory of games to include non-zero-sum games, that his theory extended the minmax theorem to such games. To invoke Nash, in a misleading manner, seems to me to be wrong—something that does not fit in print.

## Open Problems

What do you think? Not only about the NYT article, but also about how we have interpreted claims about Aryabhata. In any event, Happy Father’s Day.

June 16, 2014 1:20 am

West independently developed what mathematicians like Aryabhata and Brahmagupta had developed and perhaps that is why perhaps they are not credited for their work by West. I think that is not the case about NYT article.

• June 23, 2014 5:46 pm

No, that is not the case. The rules for algebra and place value system were mostly developed in India, and later in modern day Iraq and Iran. The translations and re-renderings of these Arabic works into Latin (most famously by Fibonacci) were crucial in setting off mathematical development in Europe. That legacy persists even now, most notoriously in the word sine for the trigonometric function, which is a Latin translation of the Arabic *transliteration* of Arayabhata’s term for arcs.

June 16, 2014 1:26 am

Well, that’s not the *whole* claim to fame for Nash. He also came up with the Nash embedding theorem, after all…not that there’s any way to use it to excuse the NYT’s mistake.

• June 17, 2014 6:26 am

Yah Nash is famous in real AG and PDEs. He probably did GT for fun like Manin did coding theory for fun.

3. June 16, 2014 6:33 am

Ignacio Palacios-Huerta/NYT observes that “The efficient-market hypothesis […] posits that the market incorporates information so completely and so quickly that any relevant news is integrated into a stock’s price before anyone has a chance to act on it.”

Gödel’s Lost Letter readers may enjoy contemplating two recent surveys of the peculiarly flimsy evidence for the efficient-market hypothesis:

Hilarity  Julian Gough’s The Great Hargeisa Goat Bubble

Sobriety  Bailey, Borwein, Lopez de Prado, and Zhu, “Pseudo-Mathematics and Financial Charlatanism: The Effects of Backtest Overfitting on Out-of-Sample Performance” (Notices of the American Mathematical Society, May 2014). The associated web site “Mathematicians Against Fraudulent Financial and Investment Advice (MAFFIA)” is plenty of fun too.

Observation  It’s exceedingly difficult — mathematically, physically, economically, and ideologically — for human beings to accept that elegantly simple cognitive frameworks can be just plain wrong.

4. June 16, 2014 10:37 am

Nash made key contributions in the theory of games, period. the exact details are not likely to be able to be communicated perfectly in popular science articles. popular science writers deserve some credit, esp those with academic credentials, they are somewhat rare, it is a hard job to do well, and the professional scientists always find some technicality about their writing incorrect. yes they are likely to leverage off the rare figures who have some public notoriety and others that are more obscure may not gain mention. this is more the nature of history & human nature than the fault of individual writers. the details are in the history books for the purists. do you really think the exact same problem could be analyzed entirely with the same result by von Neumann theory? if so, maybe theres your next blog right there!

June 16, 2014 4:52 pm

Interesting that the poster with an Indian name (Nachiketa) commented about Aryabhata and Brahmagupta — whereas the other posters (based in Europe and N.America) commented about Nash and Von Neumann 🙂

In India, Hinduism didn’t inhibit Science — it even encouraged Science. Whereas in Europe, the Church tried its best to destroy Science.

• June 16, 2014 11:15 pm

There is much disagreement on your last point. The case of Galileo is most cited, but see the very end of our “STOC 1500” post, https://rjlipton.wordpress.com/2014/05/23/stoc-1500/, or works by James Hannam, fr instance.

• June 17, 2014 2:21 pm

Some of our most profound mistakes are mistakes we make with classification scheming. One might qualify the “Church”, for example, to say that, Catholics suppressed science and Protestants encouraged it,’ but we have the same problem because all anyone will ever agree-on regarding these terms is that a pretty robust argument can be sustained by assuming that everyone agrees about what “science” means. Because you “…cannot prove a vague theory wrong (*op cit*)”, we might just-as-well propose that, education inhibits science,’ or qualify that, ‘high school coaches inhibit science education (except when teaching how to inflate soccer balls),’ etc. In the comment about “western” or “eastern” inclinations, as another example, we might therefore expect the same type of classification collision. Notice that the subject and predicate in this case, however, happen to enjoy being anchored in a classification that is not usually questioned (e.g., the name of the poster, the name of an individual about whom they commented, etc.), which is how this very thread begins using the name of the mathematician who, “is often called”.

When the name and the identity of the name agree then we can return to the important question regarding which name named the name that is the name that should be called a name in the first place! “The name of the song is called “Haddocks’ Eyes.”‘

6. June 16, 2014 10:25 pm

India had a 0 for so long and what did they do with it? Pretty much nought. Arabs got the nought and still had nought to show. The west took 0 and had nought to show for a while but after renaissance look what they have done. I really wonder when chimapnzees do arithmetic they know the 0 or not and I guess they know. Even Euclid came close to saying the boundary of a boundary is a nought but for whatever reason nought had no place value in their society.

7. August 11, 2015 4:48 pm

Brahmagupta waded into trouble only when he ruled that zero divided by zero is zero, and tried to make unique sense out of other fractions with zero.

Does this mean that he made a mistake along the way, or does this just mean that “the free commutative regular ring (for a finite set of generators)” is more complicated than the quotient of two polynomials with integer coefficients?

It is sometimes convenient define 0/0=0=1/0, as I admitted near the end of a blog post on algebraic characterizations and theorem provers:

I find commutative regular rings useful, because having a total multiplicative inverse operation seems convenient, and I find it canonical to define it the way as implied by the relation to inverse semigroups. This turns Boolean algebras into a special case of commutative regular rings, the idempotents of a strongly regular ring are a Boolean algebra, and $x\to xx^{-1}$ is the canonical projection into this Boolean algebra.

Also, this perspective suggested a formulation of the Schwartz-Zippel lemma for commutative rings instead of fields, which turned out to be valid (even so I doubt that it is more useful than the original lemma).

• August 11, 2015 5:00 pm

Now the link to the paper with a description of the free commutative regular ring should work:

Does this mean that he made a mistake along the way, or does this just mean that “the free commutative regular ring (for a finite set of generators)” is more complicated than the quotient of two polynomials with integer coefficients?